Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs

TL;DR

This work introduces a bold 3D ${ m N}=4$ rank-0 SCFTs/tensor category correspondence: each rank-0 theory is paired with a pair of non-unitary TFTs, TFT$_ ext{±}$, emergent in degenerate R-symmetry limits. The central mechanism is that supersymmetric partition functions simplify in these limits and become equal to TFT partition functions on the same manifold, with the modular data of the TFTs encoding SCFT observables. A key result is a universal lower bound on the round three-sphere free energy, $F \, ext{≥}\, - ext{log}igl( ext{sqrt}rac{5- oot 5rom 5}{10}igr)$, saturated by the minimal ${ m N}=4$ theory whose TFTs are Lee–Yang; this bound is derived from a dictionary linking $F$ to the S-matrix of the associated TFT. The paper provides explicit constructions and checks for infinitely many rank-0 theories, mapping their degenerate-limit data to known non-unitary TQFTs and exploring rich dualities among rank-0 theories, with broader implications for 3D dualities and the classification of non-unitary topological phases.

Abstract

We propose a novel procedure of assigning a pair of non-unitary topological quantum field theories (TQFTs), TFT$_\pm [\mathcal{T}_{\rm rank \;0}]$, to a (2+1)D interacting $\mathcal{N}=4$ superconformal field theory (SCFT) $\mathcal{T}_{\rm rank \;0}$ of rank 0, i.e. having no Coulomb and Higgs branches. The topological theories arise from particular degenerate limits of the SCFT. Modular data of the non-unitary TQFTs are extracted from the supersymmetric partition functions in the degenerate limits. As a non-trivial dictionary, we propose that $F = \max_α\left(- \log |S^{(+)}_{0α}| \right) = \max_α\left(- \log |S^{(-)}_{0α}|\right)$, where $F$ is the round three-sphere free energy of $\mathcal{T}_{\rm rank \;0 }$ and $S^{(\pm)}_{0α}$ is the first column in the modular S-matrix of TFT$_\pm$. From the dictionary, we derive the lower bound on $F$, $F \geq -\log \left(\sqrt{\frac{5-\sqrt{5}}{10}} \right) \simeq 0.642965$, which holds for any rank 0 SCFT. The bound is saturated by the minimal $\mathcal{N}=4$ SCFT proposed by Gang-Yamazaki, whose associated topological theories are both the Lee-Yang TQFT. We explicitly work out the (rank 0 SCFT)/(non-unitary TQFTs) correspondence for infinitely many examples.

Figures (6)

• Figure 1: A contour for the evaluation of \ref{['U(1)_1+H round shpere']}. There is a simple pole at $Z = \pi i$ inside the contour.
• Figure 2: A contour for the evaluation of \ref{['T[SU(2)] b=1 nu=0 partition function']}. Assuming $|\text{Im}[X_1]| < \pi$, there are two simple poles at $Z = \pm X_1 + \pi i$ inside the path.
• Figure 3: A path $l$ for \ref{['Simultaneous contour 1']}. There is a simple pole at $A = \pi i$.
• Figure 4: A path $l'$ for \ref{['Simultaneous Contour 2']}. There are simple poles at $A=0,\pm \pi i$.
• Figure 5: Two paths $l_1$ and $l_2$ for \ref{['Diagonal contour 1']}. There are simple poles at $A = \pm \pi i$.